1. LNF Seminar’ 2009 Frascati, Italy
Oleg N. Krokhin
Modulating Wave Holography P.N.Lebedev Physical Institute, Moscow, Russia

2. Motivation
In a standard hologram scheme the spatially-coherent radiation with a monochromatically-modulated amplitude should have a “carrier” wave interference pattern, which is to be superimposed by the pattern of the modulating wave. Respectively, image reconstruction may be performed by a frequency-modulating field. The problem is relevant to the case of distant frequencies (carrier and modulating), because the nature of interaction of electromagnetic waves with an object (“optical” properties of a medium) will be entirely different. Example: optical (modulating) and X-ray (carrier) radiation. Here arises another question: may be one can make a scheme with noncoherent carrier wave?

3. Part I
Simple variants of phase surface. Below to the right: an object consisting of two distant point sources of light.

4. Holographic method: writing of the converging spherical wave.
Question: Is the image in the focal plane defined by the field structure in any plane behind the lens?
Answer: Yes!
Practical question: Is it possible to record this structure?
Answer: Yes! One should “stop” the wave and place a sensitive photoemulsion behind the lens.
Blackened surfaces are lens equivalent.

5. Holographic method
Image reconstruction.

6. Imaging by the diffraction scheme

7. Scheme of holographic writing.
L - coherent source (laser), O - object,
L - coherent source (laser), O - object,
L - coherent source (laser), O - object,
- restored wave.
L - coherent source (laser), O - object,
L - coherent source (laser), O - object,
L - coherent source (laser), O - object,
L - coherent source (laser), O - object,
L - coherent source (laser), O - object,
L - coherent source (laser), O - object,
L - coherent source (laser), O - object,
- restored wave.
- restored wave.
- restored wave.
- restored wave.
- restored wave.
- restored wave.

8. Is it possible to make the holography scheme by the following principle: the recording is performed at one wavelength, the imaging is performed at an essentially different (modulating) wavelength?
u = a cos(Ω ∙ t – K ∙ z)∙cos(ω0 ∙ t – k0 ∙ z) = 1/2 {cos(ω2 ∙ t – k2 ∙ z) + cos(ω1 ∙ t – k1 ∙ z)} ω2 = ω0 + Ω; ω1 = ω0 – Ω

9. Standing wave in a simple case of plane counter-propagating waves u+ + u– = a {cos(Ωt – Kz) cos(ω0t – k0z) + + cos(Ωt + Kz) cos(ω0t + k0z)} = = 2a cos ω1t cos k1z + 2a cos ω2t cos k2z (u+ + u–)2 = 4a2{cos2 ω1t cos2 k1z + cos2 ω2t cos2 k2z + 2 cos ω1t cos ω2t cos k1z cos k2z}

10. A picture of a standing modulated wave in a square-law detector
Here must take place the reflection (diffraction, interference) of the reference wave at the frequency  : the image reconstruction

11. Part II
Is it possible to use a noncoherent broadband radiation as the carrier wave?
Should one modulate the signal wave intensity by a sinusoidal signal, i.e. to assume that
I = I0 + a cos (Ωt – Kz),
then, in case of a square-law detector the problem is reduced to the previous standard scheme.

12. The reconstruction may be realized by the reference wave at a frequency . The frequency of a signal radiation is of no significance. This may be an UV, soft or hard X-ray radiation.
Such a scheme gives a possibility to obtain a hologram at an optical frequency , if an object is irradiated, for example, by a more hard radiation.

13. What is the picture observed at the reconstruction?
Absolutely the same as shown in Fig. 3.
It is necessary that the object points give a sufficient scattering angle, and, then, an amplitude-phase “portrait” of the wave at a frequency  will be registered in the hologram plane.
As a square-law detector one can, obviously, use a photo-emulsion with the transmission coefficient
T  I2 .

14. CONCLUSIONS
The proposed holographic technique essentially extends the application of this method of recording, first of all in research and application tasks, particularly, for the purposes of interscopy or registration of 3D graphic presentations, for example, in medicine.